Problem 5 - A Gentle Introduction to the Art of Mathematics

Question

The student government at Lagrange High consists of 24 members chosen from amongst the general student body of \(210 .\) Additionally, there is a steering committee of 5 members chosen from amongst those in student government. Use the multiplication rule to determine two different formulas for the total number of possible governance structures.

Step-by-Step Solution

Answer

\(\binom{210}{24} \times \binom{24}{5}\) or \(\binom{210}{5} \times \binom{205}{19}\)

Step 1: Select Members of Student Government

First, we need to choose 24 members from the general student body of 210. This can be calculated using the combination formula \(_{n}C_{k}\), which represents the number of ways to choose k elements from a set of n elements without regard to the order. Therefore, we use \(_{210}C_{24}\).

Step 2: Calculate Combination for Student Government

Using the combination formula, the number of ways to select 24 students from 210 is: \[\binom{210}{24} \]

Step 3: Select Members of the Steering Committee

Next, we need to choose 5 members from the 24 that have been selected for the student government. This is also a combination problem, calculated using \(_{24}C_{5}\).

Step 4: Calculate Combination for Steering Committee

Using the combination formula, the number of ways to select 5 students from the 24 members is: \[\binom{24}{5} \]

Step 5: Applying the Multiplication Rule

To find the total number of possible governance structures, multiply the number of ways to choose the student government by the number of ways to choose the steering committee. Thus, we have: \[\binom{210}{24} \times \binom{24}{5} \]

Step 6: Alternative Formula Derivation

Alternatively, think of it as first selecting 5 members for the steering committee from the 210, followed by choosing the remaining 19 members of the student government from the remaining 205 students. This is also valid and calculated as: \[\binom{210}{5} \times \binom{205}{19} \]

Key Concepts

combination formula

In combinatorics, the combination formula is a key concept used to determine the number of ways to choose items from a group without considering the order. The combination formula is given by \(\binom{n}{k}\), which is read as 'n choose k'. This formula calculates how many ways you can pick k items from a total of n items. Mathematically, it is expressed as: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!}\] where \( n! \) (n factorial) is the product of all positive integers up to n. For example, selecting 3 students from a class of 10 can be calculated using this formula. It helps simplify problems where the order of selection doesn't matter. In our exercise, we used the combination formula to pick 24 members from 210 students and later 5 members from those 24.

multiplication rule in probability

The multiplication rule in probability is used to find the probability of two or more events happening together. If you need to determine how two independent events occur, you multiply their probabilities. Mathematically, this rule is represented as: \( P(A \text{ and } B) = P(A) \times P(B) \). In combinatorics, we use a similar concept to determine the total number of ways multiple events can occur together. In our problem, the first event is choosing student government members from the general student body, and the second event is selecting the steering committee from the chosen members. To find the total number of possible structures, you multiply the combinations for each event: \[ \binom{210}{24} \times \binom{24}{5} \]. This product gives the complete number of governance structures possible.

student governance committee

A student governance committee is typically a body of students elected or chosen to represent the student population in school matters. These committees are crucial in fostering leadership skills and promoting student involvement in school governance. In the given problem, we dealt with a student government of 24 members from a larger group of 210 students. From these 24 members, a smaller steering committee of 5 members is chosen. Understanding the combinatorial methods to form these committees helps illustrate practical applications of mathematical concepts in organizing and structuring student bodies, enhancing their functionality and representation.